You usually don't have to worry about your HQ's getting hit on the Eastern Front until later in the war and if you are in trouble as the axis. However, here are some steps to take to protect your HQ's and the axis homelands.
1. Have the HQ's accompanied by AA unit.
2. Put the HQ's as much as possible in Cities, fortifications, or mountain terrain. Avoid marshes, clear terrain, and coast lines as much as possible. HQ's are sitting ducks on a coast line for aircraft carriers.
3. Probably, by some time in 1941, start putting research into Axis AA guns. You might do it sooner than that depending on your opponent.
4. I usually will send an AA gun to North Africa to protect Rommel if I haven't pushed allies out of Africa sooner in the war.
5. Try to get AA guns to level 2 or 3 by beginning of 1943. This goes for Japs and Italy by the way. The AA guns will do lots of damage to allied airpower and are cheaper way to go than upgrading HQ's AA ratings. They also are great defending incidentally and are cheap to buy.
6. Put one or two fighter units under the command of the HQ's at highest risk of attack.
7. Keep in mind that bombers will attack city first and then there is a chance of hitting a HQ's after the bombing raid. This is why it makes sense to put units in cities for certain defensive purposes.
8. The bad news is that the strategic bombers at level 4 or 5 will and can pretty much level most cities down to zero supply. This is not as bad as it used to be, as there is always at least 1 supply point per city even if it is leveled for supply. So the axis can still move forward in Africa for example, whereas previously this was not the case.
9. If your opponent starts showing a strategy of strategic bomber use, then I would upgrade mines, factories and some cities AA defenses.
10 Fortifications can have their air defenses upgraded!! I think this is especially important in China, as folks forget to upgrade the air defenses and then get wiped out by Jap tac and medium bombers. Fortifications can be upgrade even when in contact with enemy forces.
11. If you get to level 2 AA guns, ships will take way less damage from the maritime bombers, so keep that in mind too.
12. If you can get Malta to level 2 AA guns before the axis try to blitz the place, it will make taking the place a lot more costly to the axis.
13. If on the defensive, I will then look at upgrading the HQ's AA level.
Any way, I hope this helps,